ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. l29 



and putting — 



O O 



our variation can be rearranged into — 



Remembering the few remarks made above on the subject of variation, we see that 

 55" = o is the equivalent of the fact that S itself is either maximium or minimum. In 

 other w^ords, that — 



f[T''{&y-^^^r-Hy]^l^ 



must be either maximum or minimum. By actually w^orking out the variation of this 

 integral and equating it to zero, we can obtain a differential equation, the solution of 

 which would give the required answer — the deflection in terms of the abscissa. But 

 a special method will be given in the next chapter, by which the function y can be 

 found without the calculus of variations, by the ordinary calculus rules of finding 

 maxima or minima of functions of several variables. What we have done so far 

 was to establish a certain variational equation, and to conclude therefrom that a cer- 

 tain definite integral must be either maximum or minimum. This is really the 

 starting point of Ritz's method, as will presently be shown, at which time we shall 

 return to this problem. 



4. Hamilton's Principle. — This very powerful principle of dynamics can be 

 briefly stated as follows : Let T be the kinetic and- V the potential energy ; let the 

 positions of the system be known at the moments t and ^o- Then for all compatible 

 displacement we have 



that is, the variation of the time integral of (T — V) will vanish. For further 

 particulars see Dr. Byerly's "Variations," also Slocum's "Theory and Practice of 

 Mechanics," where an instructive problem will be found worked out in detail — how 

 to find, by Hamilton's principle, the equation of the transverse vibration of a beam ; 

 the resulting equation can be written down, almost instantly, from other considera- 

 tions, but it is important to realize that there actually exists a perfectly general 

 method by which an equation of motion can be derived from the fact that a certain 

 variation of a certain definite integral vanishes. This, by the way, is as far as Ham- 

 ilton's principle can go — to enable one to form the equation of motion, but not to sup- 

 ply its integrals. 



In our problems it will not be necessary to use Hamilton's principle directly, 

 but the principle of virtual work will be applied for finding the variational equations 

 desired. We shall, however, modify the latter to fit the case of elastic vibrations, in 

 the absence of external forces — for every virtual displacement we shall equate the 



